120 research outputs found
Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
We consider elliptic partial differential equations with diffusion
coefficients that depend affinely on countably many parameters. We study the
summability properties of polynomial expansions of the function mapping
parameter values to solutions of the PDE, considering both Taylor and Legendre
series. Our results considerably improve on previously known estimates of this
type, in particular taking into account structural features of the affine
parametrization of the coefficient. Moreover, the results carry over to more
general Jacobi polynomial expansions. We demonstrate that the new bounds are
sharp in certain model cases and we illustrate them by numerical experiments
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
Elliptic partial differential equations with diffusion coefficients of
lognormal form, that is , where is a Gaussian random field, are
considered. We study the summability properties of the Hermite
polynomial expansion of the solution in terms of the countably many scalar
parameters appearing in a given representation of . These summability
results have direct consequences on the approximation rates of best -term
truncated Hermite expansions. Our results significantly improve on the state of
the art estimates available for this problem. In particular, they take into
account the support properties of the basis functions involved in the
representation of , in addition to the size of these functions. One
interesting conclusion from our analysis is that in certain relevant cases, the
Karhunen-Lo\`eve representation of may not be the best choice concerning
the resulting sparsity and approximability of the Hermite expansion
Single Bunch Instabilities in FCC-ee
FCC-ee is a high luminosity lepton collider with a centre-of-mass energy from 91 to 365 GeV. Due to the machine parameters and pipe dimensions, collective effects due to electromagnetic fields produced by the interaction of the beam with the vacuum chamber can be one of the main limitations to the machine performance. In this frame, an impedance model is required to analyze these instabilities and to find possible solutions for their mitigation. This paper will present the contributions of specific machine components to the total impedance budget and their effects on the beam stability. Single bunch instability thresholds will be estimated in both transverse and longitudinal planes
Stable high-order randomized cubature formulae in arbitrary dimension
We propose and analyse randomized cubature formulae for the numerical
integration of functions with respect to a given probability measure
defined on a domain , in any dimension . Each
cubature formula is conceived to be exact on a given finite-dimensional
subspace of dimension , and uses pointwise
evaluations of the integrand function at
independent random points. These points are distributed according to a suitable
auxiliary probability measure that depends on . We show that, up to a
logarithmic factor, a linear proportionality between and with
dimension-independent constant ensures stability of the cubature formula with
high probability. We also prove error estimates in probability and in
expectation for any and , thus covering both preasymptotic and
asymptotic regimes. Our analysis shows that the expected cubature error decays
as times the -best approximation error of
in . On the one hand, for fixed and our cubature formula
can be seen as a variance reduction technique for a Monte Carlo estimator, and
can lead to enormous variance reduction for smooth integrand functions and
subspaces with spectral approximation properties. On the other hand, when
we let , our cubature becomes of high order with spectral
convergence. As a further contribution, we analyse also another cubature
formula whose expected error decays as times the
-best approximation error of in , and is therefore
asymptotically optimal but with constants that can be larger in the
preasymptotic regime. Finally we show that, under a more demanding (at least
quadratic) proportionality betweeen and , the weights of the cubature
are positive with high probability
Numerical analysis of the factorization method for EIT with a piecewise constant uncertain background
International audienceWe extend the factorization method for electrical impedance tomography to the case of background featuring uncertainty. We first describe the algorithm for the known but inhomogeneous backgrounds and indicate expected accuracy from the inversion method through some numerical tests. Then we develop three methodologies to apply the factorization method to the more difficult case of a piecewise constant but uncertain background. The first one is based on a recovery of the background through an optimization scheme and is well adapted to relatively low-dimensional random variables describing the background. The second one is based on a weighted combination of the indicator functions provided by the factorization method for different realizations of the random variables describing the uncertain background. We show through numerical experiments that this procedure is well suited to the case where many realizations of the measurement operators are available. The third strategy is a variant of the previous one when measurements for the inclusion-free background are available. In this case, a single pair of measurements is sufficient to achieve comparable accuracy to the deterministic case
Numerical analysis of the Factorization Method for Electrical Impedance Tomography in inhomogeneous medium
The retrieval of information on the coefficient in Electrical Impedance Tomography is a severely ill-posed problem, and often leads to inaccurate solutions. It is well known that numerical methods provide only low-resolution reconstructions. The aim of this work is to analyze the Factorization Method in the case of inhomogeneous background. We propose a numerical scheme to solve the dipole-like Neumann boundary-value problem, when the background coefficient is inhomogeneous. Several numerical tests show that the method is capable of recovering the location and the shape of the inclusions, in many cases where the diffusion coefficient is nonlinearly space-dependent. In addition, we test the numerical scheme after adding artificial noise
Sparse polynomial approximation of parametric elliptic PDEs Part I: affine coefficients
International audienceWe consider the linear elliptic equation −div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ¯ a + j≥1 y j ψ j for some parameter vector y = (y j) j≥1 ∈ U = [−1, 1] N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H 1 0 (D). We consider both Taylor series and Legendre series. Previous results [8] show that, under a uniform ellipticity assuption, for any 0 p. We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. Our analysis applies to other types of linear PDEs with similar affine parametrization of the coefficients
Coupling impedances and collective effects for FCC-ee
A very important issue for the Future Circular Collider (FCC) is represented by collective effects due to the selfinduced electromagnetic fields, which, acting back on the beam, could produce dangerous instabilities. In this paper we will focus our work on the FCC electron-positron machine: in particular we will study some important sources of wake fields, their coupling impedances and the impact on the beam dynamics. We will also discuss longitudinal and transverse instability thresholds, both for single bunch and multibunch, and indicate some ways to mitigate such instabilities
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